In this paper, we consider a factor map \(\pi : (X,d, T)\rightarrow (Y, S)\) , i.e., \(T: X\rightarrow X\)  and \(S: Y\rightarrow Y\)  are two topological dynamical systems and \(\pi : X \rightarrow Y\)  is a continuous, surjective map such that \(\pi \circ T=S \circ \pi \) . Let \(C(X, \mathbb {R})\)  denote the set of all continuous functions \(\varphi : X\rightarrow \mathbb {R}\) .
The following classical relative variational principle for pressure was obtained by Ledrappier and Walters [1]} and by Downarowicz and Serafin [2]}: For any \(\nu \in \mathcal {M}_S(Y)\)  and any potential \(\varphi \in C(X, \mathbb {R})\) ,
\(\begin{aligned}\sup _{\mu \in \mathcal {M}_T(X)} \Big \lbrace h_\mu (T, X|Y)+\int _X \varphi (x) d\mu (x): \pi \mu =\nu \Big \rbrace =\int _Y P(T, \varphi , y)d\nu (y)\end{aligned}\) 
